Example 1: Multiplication before addition
Evaluate 3 + 4 × 2.
- 1There are no brackets or exponents, so move to tier 2.
- 2Multiplication: 4 × 2 = 8.
- 3Now tier 3: 3 + 8 = 11.
Answer: 11
Arithmetic · Grades 5–8
Order of Operations: Rules, Examples and Printable Test
The order of operations decides which calculation happens first when an expression contains more than one. Most students learn it as six letters, and most students then make the same two mistakes. Learning it as three tiers avoids both.
Order of Operations formula
Brackets & exponents → × and ÷ (left to right) → + and − (left to right)
There are three tiers, not six steps. Multiplication and division sit on the same tier and are worked left to right as they appear. Addition and subtraction share the tier below and are also worked left to right. Nothing inside a tier outranks anything else on that tier.
- Tier 1
- Brackets first, then exponents and roots
- Tier 2
- Multiplication and division, equal rank, left to right
- Tier 3
- Addition and subtraction, equal rank, left to right
- Left to right
- Within a tier, work in reading order
Worked examples
Example 2: Equal rank, left to right
Evaluate 12 ÷ 4 × 3.
- 1Division and multiplication share a tier, so work left to right.
- 2First: 12 ÷ 4 = 3.
- 3Then: 3 × 3 = 9.
- 4Doing the multiplication first would give 1, which is incorrect.
Answer: 9
Example 3: Brackets and exponents together
Evaluate 2 × (3 + 1)² − 5.
- 1Brackets first: 3 + 1 = 4, giving 2 × 4² − 5.
- 2Exponent next: 4² = 16, giving 2 × 16 − 5.
- 3Tier 2: 2 × 16 = 32.
- 4Tier 3: 32 − 5 = 27.
Answer: 27
Order of Operations practice questions with answers
Work through these in order. Each answer opens to show the full method, so you can check your reasoning rather than just your final number.
- Question 1easy
Evaluate: 5 + 2 × 3
Show answer and worked solution
Answer: 11
Multiply first: 2 × 3 = 6, then 5 + 6 = 11.
- Question 2easy
Evaluate: (4 + 6) ÷ 2
Show answer and worked solution
Answer: 5
Brackets first: 4 + 6 = 10, then 10 ÷ 2 = 5.
- Question 3easy
Evaluate: 10 − 3 + 2
Show answer and worked solution
Answer: 9
Same tier, so left to right: 10 − 3 = 7, then 7 + 2 = 9.
- Question 4easy
Evaluate: 3² + 1
Show answer and worked solution
Answer: 10
Exponent first: 3² = 9, then 9 + 1 = 10.
- Question 5medium
Evaluate: 20 ÷ 5 × 2
Show answer and worked solution
Answer: 8
Equal tier, left to right: 20 ÷ 5 = 4, then 4 × 2 = 8.
- Question 6medium
Evaluate: 6 + 2 × (5 − 3)
Show answer and worked solution
Answer: 10
Brackets: 5 − 3 = 2. Then 2 × 2 = 4, and 6 + 4 = 10.
- Question 7medium
Evaluate: 4 × 3²
Show answer and worked solution
Answer: 36
Exponent before multiplication: 3² = 9, then 4 × 9 = 36.
- Question 8hard
Evaluate: (8 − 2)² ÷ 4 + 1
Show answer and worked solution
Answer: 10
Brackets: 6. Exponent: 36. Divide: 36 ÷ 4 = 9. Add: 9 + 1 = 10.
- Question 9hard
Evaluate: 18 ÷ (1 + 2) × 3
Show answer and worked solution
Answer: 18
Brackets: 3. Then left to right: 18 ÷ 3 = 6, then 6 × 3 = 18.
- Question 10hard
Evaluate: 2 + 3 × 4² − 10 ÷ 5
Show answer and worked solution
Answer: 48
Exponent: 4² = 16. Tier 2: 3 × 16 = 48 and 10 ÷ 5 = 2. Tier 3: 2 + 48 − 2 = 48.
Turn this into a printable test
Generate a order of operations test at the grade and difficulty you need, with a complete worked answer key and equivalent versions for retakes.
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Why the acronym causes the errors
Six letters read as six ranked steps, and that reading is wrong twice. It implies multiplication outranks division and that addition outranks subtraction, when in fact each of those pairs is a single tier worked in reading order. The two most common order-of-operations errors in every class are direct consequences of taking the acronym literally.
Rewriting the rule as three tiers fixes both at once and takes one lesson. Students who see 12 ÷ 4 × 3 as two tier-2 operations handled left to right stop hesitating, and the same understanding carries into algebra where expressions get longer.
Where it matters beyond arithmetic
The order of operations is not a self-contained topic; it is the grammar every later topic is written in. Evaluating an algebraic expression, substituting into a formula, and reading a calculator display all depend on it. A Grade 8 student who mis-evaluates 3x² at x = 2 as 36 rather than 12 has an order-of-operations problem, not an algebra problem.
This is why the topic is worth revisiting in short bursts rather than teaching once. Mixed expressions containing an exponent, a bracket and a division catch the misconception quickly, and they take under a minute each to mark.
Common mistakes with order of operations
✗ Doing all multiplication before any division, so 12 ÷ 4 × 3 becomes 1.
✓ Multiplication and division share a tier and are worked left to right, giving 9.
✗ Doing all addition before any subtraction.
✓ Addition and subtraction also share a tier. In 10 − 3 + 2, work left to right to get 9, not 5.
✗ Applying an exponent to the coefficient, so 4 × 3² becomes 12².
✓ The exponent belongs only to the number it sits on. Evaluate 3² = 9 first, then multiply by 4.
✗ Ignoring brackets when they contain a single operation.
✓ Brackets always come first, even when what is inside looks trivial.
Order of Operations FAQ
What is the order of operations?
Brackets and exponents first, then multiplication and division left to right, then addition and subtraction left to right.
Does multiplication always come before division?
No. They share the same rank and are worked left to right, so in 12 ÷ 4 × 3 the division happens first, giving 9.
Does addition come before subtraction?
No. They also share a rank and are worked left to right, so 10 − 3 + 2 equals 9.
What does PEMDAS stand for?
Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. The MD and AS pairs are single tiers, not separate steps.
Why do calculators sometimes give a different answer?
Calculators apply the correct order, so a difference usually means brackets were left out when the expression was typed in.